I also have a 2000 with keyless entry and have on two occasions experienced problems with the security system blocking operation of the keyless system. In my case the security system was indicating that the rear hatch was not latched (indicator light showing on the dash) even though I confirmed that the hatch switch was operating correctly and had a good connection. I could manually lock the doors; but, the security system would not arm. This problem always occurred after the car came out of storage and the battery was reconnected and it always eventually went away - but not because of me. On the third occurrence of this problem the battery died (9 years old) and since battery replacement the problem has not re occurred.
I am an ex electrical engineer and have no explanation for why battery replacement resolves the problem. However, four owners have reported similar problems with the security system which were resolved by replacing the battery. Read about the weirdness here!
Hi All, The indicator light in the instrument panel will not go off when the rear glass hatch is closed. I've checked the switch, it check fine. Shaken the harness, at least as much as I can access. Shorted the plug to the switch and grounded both wires to ground. No help. On page 23-150...
www.nsxprime.com
My case is slightly different than yours because my security system would not arm, presumably because of the false hatch indication. But, that false hatch indication was originating within the security system module (definitely not an external wiring problem). You are getting a security system armed indication; but, the security system may not be providing the permissive to the keyless system / door control. This might be a case on flaky operation caused by the dreaded electrolytic capacitor problem; but, I have not seen any reports of capacitor failures in the security system or the keyless unit.
I have no analytical explanation for why battery replacement would fix this problem, just the empirical results from 4 owners where the problem went away after the battery was replaced. If your battery is more than a couple of years old, you might want to try replacement. Access to the keyless unit isn't bad; but, getting to the security module to check stuff is an absolute pain.
If the problem does appear to be the keyless unit
@drew is kind of the resident expert on the OEM keyless system. You will need the factory electrical trouble shooting manual to check the associated wiring because the factory service manual does not cover the retrofit OEM keyless system.
Edit:
Just re read your original post and noted that you said that you have replaced the car battery. Check to make sure that the battery post clamps are secure. The OEM battery post clamps are pretty flimsy and tend to get stretched form over tightening leading to flakey connections. If the clamps and battery are good, then your problem may be the keyless unit or the interface with the security system and the door lock control unit.
@drew is probably your best source on how to diagnose the keyless module to confirm whether it has failed.